Using this helpful construct, the Feynman’s Green function can now be written in a very simple fashion
\begin{equation}\label{eqn:qftLecture13:380}
\boxed{
D_F(x) = \bra{0} T(\phi(x) \phi(0)) \ket{0}.
}
\end{equation}

Remark:

Recall that the four dimensional form of the Green’s function was
\begin{equation}\label{eqn:qftLecture13:400}
D_F = i \int \frac{d^4 p}{(2 \pi)^4} e^{-i p \cdot x} \inv{ p^2 – m^2 }.
\end{equation}
For the Feynman case, the contour that we were taking around the poles can also be accomplished by shifting the poles strategically, as sketched in fig. 2.

Interacting field theory: perturbation theory in QFT.

We perturb the Hamiltonian
\begin{equation}\label{eqn:qftLecture13:500}
H = H_0 + H_{\text{int}}
\end{equation}
where \( H_0 \) is the free Hamiltonian and \( H_{\text{int}} \) is the interaction term (the perturbation).

Normally (QM) one defines the Heisenberg operator as
\begin{equation}\label{eqn:qftLecture13:640}
O_H = e^{i H(t – t_0)} O_S e^{-i H(t – t_0)},
\end{equation}
where \( O_H \) depends on time, and \( O_S \) is defined at a fixed time \( t_0 \), usually 0.
From \ref{eqn:qftLecture13:640} we find
\begin{equation}\label{eqn:qftLecture13:660}
\ddt{O_H} = i \antisymmetric{H}{O_H}.
\end{equation}
The equivalent of \ref{eqn:qftLecture13:640} in QFT is very complicated. We’d like to develop an intermediate picture.

We will define an intermediate picture, called the “interaction representation”, which is equivalent to the Heisenberg picture with respect to \( H_0 \).